Inverse Littlewood–Offord Problems for Quasi-norms

Abstract

Given a compact star-shaped domain $K\subseteq \mathbb {R}^d$, n vectors $v_1,\ldots ,v_n \in \mathbb {R}^d$, a number $R>0$, and i.i.d. random variables $\eta _1,\dots ,\eta _n$, we study the geometric and arithmetic structure of the multi-set $V = \{v_1,\ldots ,v_n\}$ under the assumption that the concentration function

$$\begin{aligned} \sup _{x\in \mathbb {R}^d}~\mathbb {P}\Big (\sum _{j=1}^n\eta _j v_j\in x+RK\Big ) \end{aligned}$$

does not decay too fast as $n\rightarrow \infty$. This generalises the case where K is the Euclidean ball, which was previously studied in Nguyen and Vu (Adv Math 226(6):5298–5319, 2011) and Tao and Vu (Combinatorica 32(3):363–372, 2012), to the non-Euclidean settings, that is, to general norms and quasi-norms in $\mathbb {R}^d$.